Hydraulic analysis of water distribution networks is an important problem in civil engineering. A widely used approach in steadystate analysis of water distribution networks is the global gradient algorithm (GGA). However, when the GGA is applied to solve these networks, zero flows cause a computation failure. On the other hand, there are different mathematical formulations for hydraulic analysis under pressuredriven demand and leakage simulation. This paper introduces an optimization model for the hydraulic analysis of water distribution networks using a metaheuristic method called shuffled complex evolution (SCE) algorithm. In this method, applying ifthen rules in the optimization model is a simple way in handling pressuredriven demand and leakage simulation, and there is no need for an initial solution vector which must be chosen carefully in many other procedures if numerical convergence is to be achieved. The overall results indicate that the proposed method has the capability of handling various pipe networks problems without changing in model or mathematical formulation. Application of SCE in optimization model can lead to accurate solutions in pipes with zero flows. Finally, it can be concluded that the proposed method is a suitable alternative optimizer challenging other methods especially in terms of accuracy.
A water distribution network is composed of an edge set consisting of pumps, pipes, valves, and a node set consisting of reservoirs and pipe intersections [
The analysis of hydraulic networks should be treated as an optimization problem, as shown by Arora [
Arora [
Now consider the network of Figure
Schematic representation of the looped pipe network with 5 unknown nodal heads.
The cocontent optimization model is expressed as
The first eight terms of the objective function represent the energy loss in real pipes
By partially differentiating (
That is the purely headbased formulations of the network equations. So cocontent model not only minimizes the energy of flow but also preserves water balance in network. For simplicity,
Collins et al. [
In the common approaches, it is presumed that the nodal demands are always satisfied at all demand nodes, irrespective of the available HGL values at demand nodes [
Water losses via leakages constitute a major challenge to the effective operation of municipal WDS since they represent not only diminished revenue for utilities but also undermined service quality [
This study introduces the shuffled complex evolution (SCE) algorithm for the hydraulic analysis. Since the algorithm was originally developed to solve optimization problems, the hydraulic network analysis was introduced into an optimization problem (cocontent model). One advantage of the SCE algorithm is that it does not need an initial solution vector which must be chosen carefully in many other procedures if numerical convergence is to be achieved. Furthermore, application of SCE algorithm in cocontent model does not require any complicated mathematical expression and operation. In this model, pressuredriven demand and leakage can be simulated easily and there is no failure in computation in zero flow conditions.
Shuffled complex evolution (SCE) is a simple powerful and populationbased stochastic optimization algorithm that outperforms many metaheuristic algorithms in numerical singleobjective optimization problems. This method is based on a synthesis of four concepts:
In SCE method, each individual represents a feasible solution for the problem. The search within the feasible region is conducted by first dividing the set of current feasible trial solutions into several complexes, each containing equal number of trial solutions. Each complex represents a local area of the whole domain. Concurrent and independent searches within each complex are conducted until each converges to its local optimal value. Each of the complexes, which are now defined by new trial solutions, is collected into a common pool, shuffled by ranking according to their objective function value, and then further divided into new complexes. The procedure is terminated when none of the local optima found among the complexes can improve on the best current local optimum. The SCE method used the downhill simplex method to accomplish local searches. So, shuffled complex evolution tries to balance between a wide scan of a large solution space and deep search of promising locations. It depends mainly on partitioning the solution space into local communities and performing local search within these communities. Then, it shuffles these local communities to perform global search.
The steps of the procedure of SCE, as shown in Figure
Step 1: initialize problem and algorithm parameters.
Step 2: samples generation.
Step 3: rank solutions.
Step 4: partition into complexes.
Step 5: start Competitive Complex Evolution (CCE).
Step 6: shuffle complexes.
Step 7: check the stopping criterion.
SCE procedure for minimization of cocontent model.
In Step 1, the optimization problem is specified as follows:
The initial population for the DE is created arbitrarily by the following formula:
In this step, the s solutions are sorted in order of increasing criterion value, so that the first vector represents the smallest value of the objective function and the last vector indicates the largest value.
The
CCE algorithm is based on the simplex downhill search scheme and is one key component of SCE algorithm. This algorithm is presented as follows.
A subcomplex by randomly selecting
The worst solution of the subcomplex is identified and the centroid of the subcomplex without including the worst solution is computed as follows:
In this step, reflection operator is used, by reflecting the worst point through the centroid according to the following formula:
If the newly generated solution is better than the worst solution, then it is replaced by the new solution. Otherwise go to
In this step, contraction operator is applied, by computing a solution halfway between the centroid and the worst point:
A solution within the feasible space is generated randomly and the worst solution is replaced by the randomly generated solution.
Steps
The solutions in the evolved complexes into a single sample population are combined and the sample population is sorted in order of increasing criterion value and is shuffled into
In this section, Steps 3, 4, and 5 are repeated until the termination criterion is satisfied.
It should be noted that the competitive complex evolution (CCE) algorithm is required for the evolution of each complex. Each point of a complex is a potential “parent” with the ability to participate in the process of reproducing offspring. A subcomplex functions like a pair of parents. Use of a stochastic scheme to construct subcomplexes allows the parameter space to be searched more thoroughly. The idea of competitiveness is introduced in forming subcomplexes where the stronger survives better and breeds healthier offspring than the weaker. Inclusion of the competitive measure expedites the search towards promising regions.
A more detailed presentation of the SCE algorithm has been given by Duan et al. [
In this section, the hydraulic analyses for several conditions in some water distribution networks are performed. All computations were executed in MATLAB programming language environment with an Intel(R) Core(TM) 2Duo CPU P8700 @ 2.53 GHz and 4.00 GB RAM. In order to demonstrate the effectiveness of SCE compared with other methods, this study proposes the use of mass balance and energy balance in the network. The average of mass and energy balance is shown by
For Figure
To check the performance of the SCE for the minimization of cocontent model, in all examples, ten optimization runs were performed using different random initial solutions.
In this part, the verification of the above mentioned model was conducted via numerical simulation based on an extremely simplified network scheme (5 nodes and 7 pipes) schematically shown in Figure
Schematic representation of the looped pipe network used in numerical example 1.
The SCE technique is applied to solve this problem according to three cases. The bound variables were set between 90 and 100 m. The problem is also solved using the global gradient algorithm (GGA) and the results are compared with those obtained by the SCE. The best, worst, and average solutions of SCE algorithm in three cases are shown in Table
Average of mass and energy balance for numerical example 1.
SCE 

Average number of function evaluations  

Best  Worst  Mean  Std  
Number of complexes  4 





Number of iterations in inner loop  4  
Number of complexes  9 





Number of iterations in inner loop  9  
Number of complexes  10 





Number of iterations in inner loop  15  


Global gradient algorithm  Maximum accuracy 

Convergence history of numerical example 1 (form 1).
Convergence history of numerical example 1 (form 2).
Example 2 considers the symmetric network shown in Figure
Schematic representation of the looped pipe network used in numerical example 2.
The SCE parameters are set as follows: number of decision variables = 7; number of points in each complex = 15; number of complexes for case 1 = 4, case 2 = 9, and case 3 = 10; number of iterations in inner loop for case 1 = 4, case 2 = 9, and case 3 = 15. The bound variables were set between 25 and 40 m. The previous best solution for this network, when it is simulated using the Elhay algorithm, and the average solution of SCE algorithm are shown in the second and third columns of Table
Head and parameter
Node number 





1  40  40  0  0 
2  36.6813  36.6853 


3  36.6813  36.6853 


4  33.3626  33.3706 


5  33.3626  33.3706 


6  30.044  30.0559 


7  30.044  30.0559 


8  26.7253  26.7411 


Average of mass and energy balance for numerical example 2.
SCE 

Average number of function evaluations  

Best  Worst  Mean  Std  
Number of complexes  4 





Number of iterations in inner loop  4  
Number of complexes  9 





Number of iterations in inner loop  9  
Number of complexes  10 





Number of iterations in inner loop  15  


Global gradient algorithm  Maximum accuracy  Fail  
Elhay algorithm  Maximum accuracy 

Convergence history of numerical example 2 (form 1).
Convergence history of head pressure (
The simplified water distribution network shown in Figure
Schematic representation of the looped pipe network used in numerical example 3.
Todini [
Head and parameter
Node number 

SCE  3 steps  SCE  3 steps  SCE  3 steps 






 
1  140  140  140  0  0  0  0 
2  80  129.304  130.07  49.304  50.07 


3  90  132.288  132.76  42.288  42.76 

0.0041 
4  70  109.587  110.96  39.587  40.96 

0.0022 
5  80  80.000  88.54  0.000  8.54  0.058 

6  90  90.000  91.45  0.000  1.45  0.0069 

7  90  90.000  90.00  0.000  0.00 

0.1421 
8  100  88.922  90.43  −11.078  −9.57 

0.0439 
Average of mass and energy balance for numerical example 3.
SCE 

Average number of function evaluations  

Best  Worst  Mean  Std  
Number of complexes  5 





Number of iterations in inner loop  5  
Number of complexes  9 





Number of iterations in inner loop  9  
Number of complexes  10 





Number of iterations in inner loop  15  


Threestep approach [ 
Maximum accuracy 

Convergence history of numerical example 3 (form 1).
Convergence history of head pressure (
The fourth considered network is a real planned network designed for an industrial area in Apulian Town (Southern Italy). The network layout is illustrated in Figure
Hydraulic data relevant to numerical example 4.
Pipe number 



1  348.5  327 
2  955.7  290 
3  483  100 
4  400.7  290 
5  791.9  100 
6  404.4  368 
7  390.6  327 
8  482.3  100 
9  934.4  100 
10  431.3  184 
11  513.1  100 
12  428.4  184 
13  419  100 
14  1023.1  100 
15  455.1  164 
16  182.6  290 
17  221.3  290 
18  583.9  164 
19  452  229 
20  794.7  100 
21  717.7  100 
22  655.6  258 
23  165.5  100 
24  252.1  100 
25  331.5  100 
26  500  204 
27  579.9  164 
28  842.8  100 
29  792.6  100 
30  846.3  184 
31  164  258 
32  427.9  100 
33  379.2  100 
34  158.2  368 
Head and parameter
Node number 






1  10.863  26.9  33.29  0.154743 

2  17.034  24.81  31.83  0.02131 

3  14.947  21.3  27.39  0.059002 

4  14.28  17.22  25.34 

0.005033 
5  10.133  23.54  30.89  0.026184 

6  15.35  20.1  29.02  0.030898 

7  9.114  18.91  27.94  0.017147 

8  10.51  17.9  27.34 

0.003545 
9  12.182  17.85  26.35 

0.003867 
10  14.579  12.66  23.24  0.008277 

11  9.007  16.23  25.95  0.031554 

12  7.575  10.12  22.05  0.002732 

13  15.2  10.03  22.45  0.0126 

14  13.55  15.41  25.95 

0.004352 
15  9.226  14  24.17 

0.002934 
16  11.2  14.36  24.05  0.007089 

17  11.469  15.3  25.42 

0.00361 
18  10.818  18.83  28.38  0.011889 

19  14.675  19.35  28.39 

0.005097 
20  13.318  10.01  23.79  0.013059 

21  14.631  11.48  22.35 

0.004141 
22  12.012  14  25.46  0.003901 

23  10.326  10.45  20.11 

0.002979 
24  —  36.45  36.45  —  — 
Average of mass and energy balance for numerical example 4.
SCE 

Average number of function evaluations  

Best  Worst  Mean  Std  
Number of complexes  5 





Number of iterations in inner loop  5  
Number of complexes  9 





Number of iterations in inner loop  9  
Number of complexes  10 





Number of iterations in inner loop  15  


Giustolisi algorithm  Maximum accuracy 

Schematic representation of the looped pipe network used in the numerical example 4.
Convergence history of numerical example 4 (form 1).
Convergence history of headpressure (
In general, 4 different pipe networks were considered in this paper and different mathematical formulations were used for the hydraulic analysis of these networks. However, the overall results indicate that the proposed method has the capability of handling various pipe networks problems with no change in the model or mathematical formulation. Application of SCE in cocontent model can result in finding accurate solutions in pipes with zero flows and the pressuredriven demand and leakage simulation can be solved through applying ifthen rules in cocontent model. As a result, it can be concluded that the proposed method is a suitable alternative optimizer, challenging other methods especially in terms of accuracy.
The objective of the present paper was to provide an innovative approach in the analysis of the water distribution networks based on the optimization model. The cocontent model is minimized using shuffled complex evolution (SCE) algorithm. The methodology is illustrated here using four networks with different layouts. The results reveal that the proposed method has the capability to handle various pipe networks problems without changing in model or mathematical formulation. The advantage of the proposed method lies in the fact that there is no need to solve linear systems of equations, pressuredriven demand and leakage simulation are handled in a simple way, accurate solutions can be found in pipes with zero flows, and it does not need an initial solution vector which must be chosen carefully in many other procedures if numerical convergence is to be achieved. Furthermore, the proposed model does not require any complicated mathematical expression and operation. Finally, it can be concluded that the proposed method is a viable alternative optimizer that challenges other methods particularly in view of accuracy.
The authors declare that there is no conflict of interests regarding the publication of this paper.